I believe "Aufnahme" can mean admission in the sense of admission to a hospital or university. I don't think you would use "Aufnahme" for admission to a show or event, which seems to be the implication here.
At least this is how i use it. Other people might use different expressions. However, "der Eintritt ist frei" is more like a fix idiomatic expression. A drink can be "frei" or "gratis". Like you said a thing can be "gratis". But "frei" und "kostenlos" are also OK.
For "Admission costs a lot", "Der Zugang kostet viel" was accepted, but here, for "Admission was free" , "Der Zugang war kostenlos" was rejected.. I guess you have to pay if you want to use "Zugang" for "Admission".